The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum context of Wheeler-DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger's norm and a conserved Mostafazadeh's norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.

We used the monodromy method to investigate the asymptotic quasinormal modes of regular black holes based on the explicit Stokes portraits. We found that for regular black holes with spherical symmetry and a single shape function, the analytical forms of the asymptotic frequency spectrum are not universal and do not depend on the multipole number, but rather on the presence of complex singularities and the trajectory of asymptotic solutions along the Stokes lines.

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